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VAR and VEC. Using Stata. VAR: Vector Autoregression. Assumptions: y t : Stationary K-variable vector v : K constant parameters vector A j : K by K parameters matrix, j=1,…,p u t : i.i.d.( 0 , S ) Trend may be included: d t, where d is K by 1 Exogenous variables X may be added.

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var and vec

VAR and VEC

Using Stata

var vector autoregression
VAR: Vector Autoregression
  • Assumptions:
    • yt: Stationary K-variable vector
    • v: K constant parameters vector
    • Aj: K by K parameters matrix, j=1,…,p
    • ut: i.i.d.(0,S)
  • Trend may be included: dt, where d is K by 1
  • Exogenous variables X may be added
var and vec3
VAR and VEC
  • If yt is not stationary, VAR or VEC can only be applied for cointegrated yt system:
    • VAR (Vector Autoregression)
    • VEC (Vector Error Correction)
vec vector error correction
VEC: Vector Error Correction
  • If there is no trend in yt, let P = ab’ (P is K by K, a is K by r, b is K by r, r is the rank of P, 0<r<K):
vec vector error correction5
VEC: Vector Error Correction
  • No-constant or No-drift Model: v = 0 g = 0, m = 0
  • Restricted-constant Model: v = amg = 0, m≠ 0
  • Constant or Drift Model: g≠ 0, m≠ 0
vec vector error correction6
VEC: Vector Error Correction
  • If there is trend in yt
vec vector error correction7
VEC: Vector Error Correction
  • No-drift No-trend Model: v = 0, d = 0 g = 0, m = 0, t = 0, r = 0
  • Restricted-constant Model: v = am, d = 0 g = 0, m≠ 0, t = 0, r = 0
  • Constant or Drift Model: d = 0 g≠ 0, m≠ 0, t = 0, r = 0
  • Restricted-trend Model: d = arg≠ 0, m≠ 0, t = 0, r≠ 0
  • Trend Model:g≠ 0, m≠ 0, t≠ 0, r≠ 0
example
Example
  • C: Personal Consumption Expenditure
  • Y: Disposable Personal Income
  • C ~ I(1), Y ~ I(1)
  • Consumption-Income Relationship:Ct = vc + acc1Ct-1 + acy1Yt-1 +…(+ dct) + ecYt = vy + ayc1Ct-1 + ayy1Yt-1 +…(+ dyt) + ey
  • Cointegrated C and I: e ~ I(0)?
    • Which model? (trend and/or drift)
    • How many lags?
example johansen test with constant models and 6 lags11
ExampleJohansen Test with constant models and 6 lags
  • Constant model: v = am + g (a 0,g0)
  • Restricted-constant model: v = am (a 0)
  • No-constant model: v = 0
example johansen test with trend models and 6 lags12
ExampleJohansen Test with trend models and 6 lags
  • Trend model:d = ar+t g≠ 0, m≠ 0, t≠ 0, r≠ 0
  • Restricted trend model: d = ar g≠ 0, m≠ 0, t = 0, r≠ 0
  • No trend or constant model: d = 0 g≠ 0, m≠ 0, t = 0, r= 0
example vec estimated models with 6 lags
Example VEC estimated models with 6 lags
  • Empirical Model: 1948q3 – 2012q3 (N=257)
    • No-constant: LL=1766.15 (K=23)
    • Restricted-constant: LL=1771.47 (K=24)
    • Restricted-trend: LL=1774.15 (K=27)
vec forecasting estimated models 1948q3 2009q4
VEC Forecasting Estimated Models: 1948q3 – 2009q4
  • No-constant model, 6 Lags (LL=1684.37, 23 parameters, N=246)
  • Restricted-constant model, 6 lags (LL=1689.23, 24 parameters, N=246)
  • Restricted-trend model, 6 lags (LL = 1691.9, 27 parameters, N=246)